14.123 Microeconomics III– Problem Set 3
Muhamet Yildiz
Instructions. Each question is 33 points. Good Luck!
1. Let P be the set of lotteries over {a, b, c} × {L, M, R}. In which of the following pairs
of games the players’preferences over P are the same?
(a)
L
M
R
a 2,-2 1,1 -3,7
b 1,10 0,4 0,4
c -2,1 1,7 -1,-5
L
M
R
a 12,-1 5,0 -3,2
b 5,3 3,1 3,1
c -1,0 5,2 1,-2
L
M
R
a 1,2 7,0 4,-1
b 6,1 2,2 8,4
c 3,-1 9,2 5,0
L
M
R
a 1,5 7,1 4,-1
b 6,3 2,4 8,8
c 3,-1 9,5 5,1
(b)
2. Let P be the set of all lotteries p = (px , py , pz ) on a set C = {x, y, z} of consequences.
Below, you are given pairs of indifference sets on P . For each pair, check whether
the indifference sets belong to a preference relation that has a Von-Neumann and
Morgenstern representation (i.e. expected utility representation). If the answer is
Yes, provide a Von-Neumann and Morgenstern utility function; otherwise show which
Von-Neumann and Morgenstern axiom is violated. (In the figures below, setting pz =
1 − px − py , we describe P as a subset of R2 .)
(a) I1 = {p|px = 2py + 1} and I2 = {p|px = 4py + 1}
(b) I1 = {p|px = 2py + 1} and I2 = {p|px = 2py }
(c) I1 = {p|px ≤ 1/2} and I2 = {p|px > 1/2}
(d) I1 = p|py = (px )2 + 1/2 and I2 = p|py = (px )2
3. On a given set of lotteries, find a discontinuous preference relation t that satisfies the
independence axiom.
1
MIT OpenCourseWare
http://ocw.mit.edu
14.123 Microeconomic Theory III
Spring 2015
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .